In mathematical logic, suppose T is a theory in the language $L = \langle \in \rangle$ of set theory.

If $M$ is a model of $L$ describing a set theory and N is a class of M
such that $ \langle N, \in_M, \ldots \rangle$ is a model of T
containing all ordinals of M then we say that N is an inner model of T
(in M). Ordinarily these models are transitive subsets or subclasses
of the von Neumann universe V, or sometimes of a generic extension of
V. (inner model, Wikipedia)

Question is, how can N contain all ordinals of M? Shouldn't a model contain ordinals as its domain? If models have same ordinals as its domain, aren't they basically the same, not needing any concept like inner model?

What am I getting wrong about a model? What makes an inner model different?

If I am not mistaken, it seems to say that ZFC has a countable inner model, which means countable domain - countable ordinals... and this just does not make sense.

No, it does not say ZFC has a countable inner model. The requirement that the membership relation of $N$ is the restriction of the membership relation of $M$ is quite strong, and applying the Löwenheim–Skolem theorem generally does not produce models of this kind.
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Zhen LinSep 18 '12 at 7:26

2 Answers
2

You are confusing between ordinals and sets. Some sets are ordinals, not all sets are ordinals. In fact not all sets are sets of ordinals, and not all sets are sets of sets of ordinals, and so on.

To understand inner models slightly better you have to study from an actual book, not from Wikipedia. There are a lot of subtle issues which you really need to sit and prove things for yourself before you fully understand.

Now, to your question. There are two ways to look at universes of set theory. You can assume that you have one universe and you live inside this universe. Internally, this universe is not a set. We say that $M$ is an inner model of this universe if it is a transitive class containing all the ordinals, but has less sets, in which the axioms of ZF[C] hold. For example, you can define $L$ internally so you have $L$ as an inner model of every universe of ZFC.

Or, we could assume that we have sets which are models of ZFC, which is a slightly stronger assumption since this is equivalent to saying that ZFC is consistent, and we cannot prove that. We are allowed to assume it, though. So now we have set models which can be very nice and have some of the real ordinals, or they could be very bad models and don't have many the real ordinals at all. So now I have two set models, and let us assume that they are nice, the two set models may or may not have the same ordinals. They may have the same ordinals, but completely different sets as well.

We say, again, that $M$ is an inner model of $N$ if $M\subseteq N$, and $M$ contains all the ordinals of $N$, and so on. However between the ordinals of $N$ (the larger one) and the ordinals of the universe there doesn't have to be any relation whatsoever.

The domain of a set theoretical model does not just contain ordinals, it contains all the sets of the model, therefore two models with the same ordinals can differ in many ways. For example they can have different powersets $P(\alpha)$ of an ordinal $\alpha$, or the set of functions from an ordinal $\alpha$ to an ordinal $\beta$ can differ. This is the reason why the Continuum Hypothesis can not be decided within ZFC.